1. Field of the Invention
The present invention relates to heat exchange devices and, more particularly, to a heat dissipating fin associated with a heat exchange device.
2. Description of the Related Art
Whenever energy is used to perform work, part of the energy may be converted to thermal energy in accordance with the first law of thermodynamics. For example, when energy in the form of a voltage potential is used to cause electrons to move in conductive materials, such as, for example, in a transistor, part of the energy is converted to thermal energy. On a macroscopic scale, when an energy source is used to move two materials that are in contact with each other, such as a rotating shaft on a bushing, the irregularities on the surface of the two materials interact, causing friction and the conversion of some of the source energy to thermal energy.
In those examples, the change in the internal energy of the system, that is, the conversion of energy to thermal energy, reflects the fact that systems are not one-hundred percent efficient. To maximize the efficiency of such a system, the thermal energy by-product must be removed from or transferred out of the system. Therefore, the characteristics of heat transfer, or heat exchange, become a crucial design element for many systems.
The term heat refers to the exchange of the thermal energy being transferred from a hot body to a cold body. When the hot body and the cold body come in contact with each other, the heat will flow from the hot body to the cold body until they both reach the same temperature (i.e., thermal equilibrium). Heat is capable of being transferred through solid and fluid media by conduction, through fluid media by convection, and through vacuum by radiation. The challenge in designing a heat exchange device that takes advantage of those heat transfer mechanisms is to design one that balances efficiency with economics. Often, the most efficient heat exchange devices are expensive to manufacture and operate. Less expensive devices may not achieve the desired heat transfer efficiency.
Fins are surface extensions frequently used in heat exchange devices for the purpose of increasing heat transfer rates, and hence overall heat transfer efficiency, between a hot body (e.g., a solid surface at a high temperature) and a cold body (e.g., a fluid surrounding the solid surface at a lower temperature). With fins, heat will flow from the high temperature solid surface (source) to the lower temperature fluid surrounding the solid surface (sink) so that eventually a constant temperature difference between the surface and the fluid (i.e., dynamic thermal equilibrium) will be reached. Heat transfer efficiency can be increased further through forced convection by using fans or pumps to move the fluid relative to the solid surface.
In many applications, fins require no maintenance. Therefore, operating costs associated with those fins are essentially negligible. Thus, in addition to increasing heat transfer rates, fins are economically attractive for use in certain systems.
Theoretical and experimental studies have suggested that the thermal efficiency of fins depends on their shape. Fins with variable thickness were first considered in Mathematical Equations for Heat Conduction in the Fins of Air Cooled Engines by D. R. Harper and W. B. Brown (NACA Report No. 158 (1922)) and in Die Wärmeübertragung durch Rippen, by E. Schmidt (Z. Ver. Deut. Ingenieure 70, at 885-889, 947-951 (1926)). The problem addressed by those references consisted of minimizing the volume of a straight symmetric fin that dissipates a given amount of heat at a given temperature difference to the ambient fluid. In the former reference, the problem of finding an optimal fin design was not addressed. In the later reference, the problem was solved using the “length of arc” assumption.
From a physical standpoint, the “length of arc” assumption is equivalent to the assumption that heat is dissipated from the fin to the surrounding fluid in the direction orthogonal to the plane of symmetry of the fin (i.e., in the y-axis direction as shown in FIGS. 1-6). In reality, however, the direction of heat flux from the fin to the fluid is orthogonal to the fin surface. Thus, in the case of the triangular-shaped fin illustrated in FIG. 2, for example, the heat transfer is dissipated out from the sides of the fin at an angle 90°-α, relative to the x-axis shown.
In Schmidt's study, with the “length of arc” assumption employed, it was found that the best dimensions for a straight fin are those that produce a linear temperature profile along the length of the fin (from its base to its tip). In that situation, the heat flux along the fin is uniform (the validity of Schmidt's analysis was confirmed mathematically in A Variational Problem Relating to Cooling Fins, by R. J. Duffin (J. Math. Mech. 8:47-56 (1959))). Proceeding from the linear temperature distribution along the fin length, Schmidt found that the optimum profile of a straight fin is convex parabolic, as illustrated in FIG. 3. The shape of the parabola in FIG. 3 is determined by two pre-specified quantities of thermal nature: the ratio, γ, of the heat transfer coefficient, h, between the fin and the fluid and the thermal conductivity, k, of the fin material, i.e., γ=h/k, and the dimensionless quantity ρ=qo/(k θ0), where qo is the heat flow through the fin semi-base per unit depth and θ0 is the difference between the temperatures of the heated surface and the surrounding fluid.
A key assumption shared by later practitioners, as discussed to some extent in Heat Transfer, by M. Jakob (10th ed., vol. 1, Wiley, New York, pp. 217-221 (1967)), is the omission of the curvature of the fin profile from the analysis of fins (i.e., omission of the “length of arc” assumption). In The Minimum Weight One-Dimensional Straight Cooling Fin, by C. J. Maday (ASME J. Eng. Ind. 96:161-165 (1974)), the impact of the “length of arc” assumption is analyzed with regard to the optimum profile shape of straight fins. It points out that the differential area element of the semi-surface per unit depth applicable to the straight fin should be expressed by the relation dS=[(1+(y′)2)1/2]dx, where y=y(x) stands for the fin profile function (see FIG. 3). In particular, the approximation dS≈dx is equivalent to the “length of arc” assumption.
The problem of finding an optimum fin design has been centered on a search for the optimum fin length L, fin semi-height at the fin base, yo, and the fin profile y=y(x), 0≦x≦L, y(0)=yo (see FIG. 3), given the thermal parameters qo, θ0, h and k. In contrast to this, the goal established in The Minimum Weight One-Dimensional Straight Cooling Fin was to find a minimum volume fin with a fixed yo. The two-point boundary value problem was solved numerically using the Pontryagin's Maximum Principle. The optimum profile reported is reasonably close to Schmidt's convex parabolic profile for a large initial portion of the fin length, but closer to the end contains some wavy irregularities. In addition, the volume of the fin was only slightly smaller than the volume of Schmidt's fin with the same height. An important numerical finding was that, with the “length of arc” assumption omitted, the temperature distribution for the optimum fin was still linear.
Thus, before the present invention, the exact profile of a straight, solid heat exchange fin that minimizes the fin volume and produces dissipation of a given heat flow per unit depth to the surrounding fluid at a given temperature excess at the fin base had not been known. Previous theoretical heat exchange devices relied on imperfect fin profiles to transfer heat, thus limiting the thermal efficiency of the devices. Moreover, those previous fin shapes were selected for economical reasons as much as for efficiency. For example, a rectangular-shaped fin shown in FIG. 1, with dimensions L (length)×W (width)×H (height), is relatively easy to manufacture, but is inefficient from a thermal performance perspective. Thermal characteristics of a rectangular-shaped fin are presented in Mathematical Equations for Heat Conduction in the Fins of Air Cooled Engines, by D. R. Harper and W. B. Brown (NACA Report No. 158 (1922)).
It is well known that to economize on the weight of the fin, it should taper approximately to a point in the direction of heat flow. Thus, triangular-shaped fins, as shown in FIG. 2, would be preferable over rectangular-shaped fins. In one aspect, triangular-shaped fins cost less than rectangular fins because they use less material, but overall they can be more expensive to manufacture because of the angled surfaces. Nevertheless, triangular-shaped fins are more efficient than rectangular-shaped fins and are often used in heat exchange devices. Thermal characteristics of a triangular-shaped fin are presented in Mathematical Analysis of the Length-of-Arc Assumption, by S. Graff and A. D. Snider (Heat Transfer Eng. 8 (2):67-71 (1996)).
The convex-parabolic-shaped fin shown in FIG. 3 is even more efficient than the rectangular- and triangular-shaped fins, but is more expensive to manufacture because of the curved sides of the fin and because the overall size of the fin requires more material to make it. Thermal characteristics of the convex-parabolic-shaped fin design is discussed, to some extent, in Heat Transfer and in Mathematical Analysis of the Length-of-Arc Assumption cited above.
Another fin design is one that is semi-rectangular shaped, as shown in FIG. 4. Thermal characteristics of a semi-rectangular shaped fin are described in Determination of the Optimum Profile of One-Dimensional Cooling Fins, by S. K. Hati and S. S. Rao (ASME J. Vib. Acoust. Stress Reliab. Des. 105:317-320 (1983)). That publication discloses using a numerical technique to find that the optimum fin profile has a depth that gradually increases toward the middle of the fin length and then decreases continuously. About two-thirds of the total heat is transferred to the surroundings from the first half of the length of the fin.
Straight, curved, and convex fins are disclosed in U.S. Pat. No. 4,669,685 to Dalby for use in dissipating heat generated by systems enclosed in a satellite. Trapezoidal and rectangular-shaped fins are disclosed in U.S. Patent Appl. Publication No. 2002/0074114-A1 to Fijas for use in dissipating heat in a fin-tubed heat exchanger. U.S. Pat. No. 5,729,988 to Tchernev illustrates how a straight, triangular-shaped fin is used to dissipate heat in a heat pump system. U.S. Pat. No. 6,161,610 to Azar discloses arc-shaped (half-round) fins (non-solid).
Until the present invention, heat exchange fins were not optimal in terms of heat transfer efficiency. In contrast to previous fins, the present invention minimizes the fin volume and at the same time procures the dissipation of a given heat flow per unit depth at a given temperature difference to the surrounding fluid and takes full account of the curvature of the fin surface.